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Infrared divergences have long been heralded to cancel in sufficiently inclusive crosssections, according to the famous Kinoshita-Lee-Nauenberg theorem which mandates an initial and final state sum. While well-motivated, this theorem is much weaker than necessary: for finiteness, one need only sum over initial or final states. Moreover, the cancellation generically requires the inclusion of the forward scattering process. We provide a number of examples showing the importance of this revised understanding: in e + e − → Z at next-to-leading order, one can sum over certain initial and final states with an arbitrary number of extra photons, or only over final states with a finite number of photons, if forward scattering is included. For Compton scattering, infrared finiteness requires the indistinguishability of hard forward-scattered electrons and photons. This implies that in addition to experimental limits on the energy and angular resolution, there must also be an experimental limit on the momentum at which electric charge can be observed. Similar considerations are required to explain why the rate for γγ to scatter into photons alone is infrared divergent but the rate for γγ to scatter into photons or charged particles is finite. This new understanding sheds light on the importance of including degenerate initial states in physical predictions, the relevance of disconnected Feynman diagrams, the importance of dressing initial or final-state charged particles, and the quest to properly define the S matrix.

We use the one-dimensional object-oriented particle-in-cell Monte Carlo collision code oopd1 to explore the evolution of the charged particle density profiles, electron heating mechanism, the electron energy probability function (EEPF), and the ion energy distribution in a single frequency capacitively coupled oxygen discharge, with driving frequency in the range 12-100 MHz. At a low driving frequency and low pressure (5 and 10 mTorr), a combination of stochastic (α-mode) and drift ambipolar (DA) heating in the bulk plasma (the electronegative core) is observed and the DA-mode dominates the time averaged electron heating. As the driving frequency or pressure are increased, the heating mode transitions into a pure α-mode, where electron heating in the sheath region dominates. At low pressure (5 and 10 mTorr), this transition coincides with a sharp decrease in electronegativity. At low pressure and low driving frequency, the EEPF is concave. As the driving frequency is increased, the number of low energy electrons increases and the relative number of higher energy electrons (>10 eV) increases. At high driving frequency, the EEPF develops a convex shape or becomes bi-Maxwellian.

We generalize the relation between discontinuities of scattering amplitudes and cut diagrams to cover sequential discontinuities (discontinuities of discontinuities) in arbitrary momentum channels. The new relations are derived using time-ordered perturbation theory, and hold at phase-space points where all cut momentum channels are simultaneously accessible. As part of this analysis, we explain how to compute sequential discontinuities as monodromies and explore the use of the monodromy group in characterizing the analytic properties of Feynman integrals. We carry out a number of cross-checks of our new formulas in polylogarithmic examples, in some cases to all loop orders.

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